Methods and apparatus for determining shape parameter(s) using a sensing fiber having a single core with multiple light propagating modes

ABSTRACT

Example embodiments include an optical interrogation system with a sensing fiber having a single core, the single core having multiple light propagating modes. Interferometric apparatus probes the single core multimode sensing fiber over a range of predetermined wavelengths and detects measurement interferometric data associated with the multiple light propagating modes of the single core for each predetermined wavelength in the range. Data processing circuitry processes the measurement interferometric data associated with the multiple light propagating modes of the single core to determine one or more shape-sensing parameters of the sensing fiber from which the shape of the fiber in three dimensions can be determined.

This application is a continuation of and claims the benefit of priorityunder 35 U.S.C. § 120 to U.S. patent application Ser. No. 16/474,531,filed on Jun. 27, 2019, which is a U.S. National Stage Filing under 35U.S.C. 371 from International Application No. PCT/US2017/067588, filedon Dec. 20, 2017, and published as WO 2018/125713 A1 on Jul. 5, 2018,which claims the priority and benefit of U.S. Provisional PatentApplication 62/440,035, filed Dec. 29, 2016, entitled “METHODS ANDAPPARATUS FOR DETERMINING SHAPE PARAMETER(S) USING A SENSING FIBERHAVING A SINGLE CORE WITH MULTIPLE LIGHT PROPAGATING MODES,” each ofwhich is incorporated herein by reference in its entirety.

TECHNICAL FIELD

The technology described in this application relates to opticalinterrogation system measurements used for fiber optic shape and othersensing applications.

INTRODUCTION

Optical strain sensing is a technology useful for measuring physicaldeformation of a waveguide caused by, for example, the change intension, compression, or temperature of an optical fiber. Measuring theshape of an optical fiber using multiple single mode cores to determineparameters including pitch, yaw, twist, and strain has beendemonstrated. Shape is the position or orientation of the fiber in threedimensions. A continuous measure of strain along the length of a corecan be derived by interpreting the optical response of the core usingswept wavelength inteferometery typically in the form of OpticalFrequency Domain Reflectometry (OFDR) measurements. With knowledge ofthe relative positions of the cores along the length of the fiber, theseindependent strain signals may be combined to gain a measure of thestrain profile applied to the multi-core optical fiber. The strainprofile of the fiber refers to the measure of applied bend strain, twiststrain, and/or axial strain along the length of the fiber at a high(e.g., less than 50 micrometers) sample resolution. A non-limitingexample application is robotic arms used in surgical or otherenvironments. Previous patents have described OFDR-based shape sensingwith multiple single mode cores (e.g., see U.S. Pat. Nos. 7,781,724 and8,773,650 incorporated by reference).

Fibers with multiple single mode cores are difficult and costly tomanufacture. Cost is further increased when multi-core fibers must bespun or helically-twisted during manufacture, which is the case forprior multi-core shape sensing fiber. If a single core, unspun fibercould be used to sense shape, then the cost of the fiber sensor could besignificantly reduced. For example, mass produced, standard telecomfiber that has single cores and is unspun sells for pennies per meter.But there are several technological challenges (described below) thatmust be addressed and overcome in order for a single core, multiple modefiber to sense shape.

SUMMARY

Example embodiments include an optical interrogation system and methodthat includes a sensing fiber having a single core, where the singlecore has multiple light propagating modes. The system includesinterferometric apparatus that probes the single core over a range ofpredetermined wavelengths and detects measurement interferometric dataassociated with the multiple light propagating modes of the single corefor each predetermined wavelength in the range. Data processingcircuitry processes the measurement interferometric data associated withthe multiple light propagating modes of the single core to determine ashape-sensing parameter of the sensing fiber. In an example application,the data processing circuitry determines a shape of the sensing fiberbased on the shape-sensing parameter.

Examples of a shape-sensing parameter include a bend parameter and/or atwist parameter. In one example embodiment, the data processingcircuitry processes the measurement interferometric data associated withthe multiple light propagating modes of the single core to determine aplurality of shape-sensing parameters including the shape-sensingparameter, and wherein the plurality of shape-sensing parametersincludes strain, bend, and twist parameters.

The single core may have a shape that limits a number of the multiplelight propagating modes below a predetermined number while providing apredetermined sensitivity to the twist parameter. One example is wherethe single core is ring-shaped in cross-section. An even more specificring-shaped core example is a ring-shaped cored that has a radius thatpermits fewer than 40 modes of light to propagate along the single core.Another example is where the single core is a solid core. A radius ofthe solid core permits less than six modes of light to propagate alongthe single core.

In example embodiments, the interferometric apparatus may includemultiple interferometers with multiple reference branches and multiplemeasurement branches having an array of corresponding multiple singlecore, single mode fibers. A tunable laser generates light over the rangeof predetermined wavelengths. Each fiber of the array has a differentoptical delay. A collimator collimates light from the single core andde-collimate light to the single core, and a microlens array receivesand focuses collimated light from the collimator onto the array ofcorresponding multiple single core, single mode fibers. The differentoptical delays produce multiple coupling coefficients that appear ondifferent ones of the array of corresponding multiple single core,single mode fibers.

In an example implementation, the single core includes an overlappinggrating pattern, where each overlapping grating in the overlappinggrating pattern is tilted with respect to a longitudinal axis of thesingle core. The overlapping grating pattern varies with bend, strain,and twist applied to the sensing fiber. The overlapping grating patternis associated with (i) a cross-sectional index perturbation for thesensing fiber as a function of distance along the sensing fiber and (ii)coupling coefficients between back-scattered light propagating modes forthe sensing fiber.

In an example application, the interferometric apparatus measures aphase and an amplitude of the coupling coefficients, and the dataprocessing circuitry determines a difference between the measuredcoupling coefficient phase and amplitude and a predetermined baselinecoupling coefficient phase and amplitude for the sensing fiber.

In an example embodiment, a non-transitory machine-readable mediumstores a plurality of machine-readable instructions which, when executedby one or more processors associated with a medical device, are adaptedto cause the one or more processors to perform processing steps of themethods described herein.

BRIEF DESCRIPTION OF THE FIGURES

FIGS. 1A and 1B show a fiber with two single-mode cores displaced fromthe central axis and a Bragg grating in an unbent and bent shape.

FIG. 2 illustrates first and second order mode fields in a twodimensional waveguide.

FIG. 3 shows an example of a sum and a difference of first and secondorder modes in a two dimensional.

FIGS. 4A and 4B show straight and bent conditions of a two dimensionalwaveguide and how it affects the spacing in a Bragg grating in thewaveguide.

FIG. 5 illustrates coupling of a forward propagating mode to a backwardpropagating mode using a Bragg grating.

FIGS. 6A-6D show all of the forward to backward coupling modes in theexample from FIG. 5.

FIG. 7A shows a grating in a multi-core fiber; FIG. 7B shows the samefiber when bent; and FIG. 7C shows a large core fiber with the samegrating pattern bent in the same way.

FIG. 8 is a reference diagram of a single core fiber with axis and anglelabels.

FIG. 9 shows an example tilted Bragg grating.

FIG. 10 shows an example intensity plot of the index modulation of atilted Bragg grating.

FIG. 11 shows an example phase mask to write a Bragg grating.

FIG. 12 illustrates an example tipped phase mask to produce a tiltedBragg grating.

FIG. 13 shows an example intensity plot of the index change created by atilted grating.

FIG. 14 shows an example intensity plot of a pattern created when twotilted gratings are written overtop of one another.

FIG. 15 shows an example amplitude envelope of overlapped tiltedgratings.

FIG. 16 is an example intensity plot of a germanium doping concentrationin a fiber core.

FIG. 17 illustrates intra-core features created by writing tiltedgratings on top of one another within the optical fiber core.

FIG. 18 is an example intensity plot of an end view of the fiber withfeatures created by writing tilted gratings on top of one another in thecore.

FIG. 19 shows scalar field plots of three lowest order LP modes.

FIG. 20 shows decomposition functions for decomposing indexperturbation.

FIG. 21 shows simulated compound tilted grating index profiles and theprofiles reconstructed from the overlap function decomposition.

FIG. 22 shows an example of a bulk-optic design for coupling laser lightfrom a fiber array into a single core fiber.

FIG. 23 shows an example system for interrogating a single core,multiple mode sensing fiber.

FIG. 24 illustrates delay locations of time-domain measurements of thecoupling coefficients.

FIG. 25 shows an example model of an optical connection system as amatrix.

FIG. 26 is a graph of an example impulse response of a few-mode fiberwith a cleaved end.

FIG. 27 is a graph of an example impulse response drawing with couplingcoefficients labeled.

FIG. 28 graphs example fiber mode responses with exaggerated time-lengthdifferences.

FIG. 29 graphs example resampling and alignment of couplingcoefficients.

FIG. 30 graphs example phase accumulations for different couplingcoefficients as a function of delay.

FIG. 31 is a flowchart showing example procedures for using a singlecore, multiple mode fiber for sensing shape in accordance with exampleembodiments.

FIG. 32 is a flowchart showing example procedures for using a singlecore, multiple mode fiber for sensing shape in accordance with exampleembodiments.

FIG. 33 is a flowchart showing example procedures for calibrating andthen using a single core, multiple mode fiber for sensing shape inaccordance with example embodiments.

FIGS. 34A and 34B show a skew ray in a fiber core and an axial ray in afiber core, respectively.

FIG. 35 shows an example single core fiber with three modes.

FIG. 36 is a graph showing modal propagation values for LinearPolarization Modes vs. V-number.

FIG. 37 is a graph showing normalized propagation constant B versusnormalized frequency V for TE and TM modes.

FIG. 38 shows an example single large core fiber with 200 modes.

FIG. 39 shows an example single core fiber where the core is annular andsupports 30 modes.

DETAILED DESCRIPTION

The following description sets forth specific details, such asparticular embodiments for purposes of explanation and not limitation.But it will be appreciated by one skilled in the art that otherembodiments may be employed apart from these specific details. In someinstances, detailed descriptions of well-known methods, interfaces,circuits, components, and devices are omitted so as not to obscure thedescription with unnecessary detail. It will be appreciated by thoseskilled in the art that diagrams herein can represent conceptual viewsof illustrative circuitry, components, or other functional units.

Two Dimensional Waveguide Example

Before considering a single core fiber in three dimensions, consider afiber with two single-mode cores displaced from the central axis and aBragg grating present in the structure from two dimensions. FIGS. 1A and1B show (two-dimensionally) a fiber 8 with two single-mode cores 12, 14displaced from the central axis and a Bragg grating 16 in an unbent andbent shape, respectively. The grating forms a set of vertical planes inthe core in FIG. 1A and tilted planes in FIG. 1B. With the fiber unbent,the two cores reflect the same wavelength. When the fiber is bent, thetop core is stretched and the bottom core is compressed, which changesthe periodicity of the Bragg grating 16 seen by each core. As a result,a different wavelength is reflected at the grating in each of the twosingle-mode cores.

If there is only a single core in the fiber, but one that supports twofield propagation modes, then the two modes will have electric fieldenvelopes that look like those in FIG. 2 which illustrates a first lowerorder mode field E₀ and a second higher order mode field E₁ in a twodimensional fiber waveguide. If both modes are launched, they propagatedown the fiber with a linearly varying phase between them. When the twomodes are in phase, their fields will sum and the total [E₀+E₁] will beweighted toward the top of the waveguide, and when they are out ofphase, their fields will subtract and the combination [E₀−E₁] will beweighted toward the bottom of the waveguide as illustrated by the“peaks” shown in FIG. 3.

By using the sum and difference of the modes, reflections preferentiallyweighted in the top or the bottom of the core can be used to detectbending of the waveguide, i.e., to detect strain which is one of theshape determination parameters. FIGS. 4A and 4B show straight and bentconditions of a two dimensional fiber waveguide and how it affects thespacing in a Bragg grating in the waveguide. Bending the waveguidecauses the top and bottom of the waveguide to expand or compress. Ifthere is a grating physically in the core, then reflections from thiscore grating can be detected in both the top and the bottom of thewaveguide. In FIG. 4B, the sum of the fields will measure the grating inthe top of the core, and the difference of the fields will measure thegrating in the bottom of the core.

Rather than directly measuring the propagating modes, measuring thecross coupling of those modes in the core grating can provide ameasurement of the deformation of the grating which can be used todetermine shape parameters. Cross coupling coefficients, κ, representthe scattering of light from the forward propagating modes to thebackward propagating modes in the core grating. FIG. 5 illustrates anelectric field incident on and interacting with the grating resulting insome of the incident field E⁺ being transmitted (a forward propagatingmode) and some of the incident field E⁻ being reflected (a backwardpropagating mode). Another way of describing this interaction is “crosscoupling” (or simply “coupling”) the forward propagating mode and thebackward propagating mode in the grating.

For a two mode, two dimensional wave guide, four types of backwardcoupling can occur including: coupling from the forward propagatingfirst order mode to the backward propagating first order mode (andhaving a coupling coefficient κ₀₀), coupling from the forwardpropagating first order mode to the backward propagating second ordermode (and having a coupling coefficient κ₀₁), coupling from the forwardpropagating second order mode to the backward propagating second ordermode (and having a coupling coefficient κ₁₁), and coupling from theforward propagating second order mode to the backward propagating firstorder mode (and having a coupling coefficient κ₁₀). FIGS. 6A-6D show allof the forward to backward coupling modes in this example.

Three Dimensional Waveguide Example

Modeling waveguide systems as ideal lossless waveguide systems, couplingcoefficients κ₀₁=κ*₁₀, and so there are three independent couplingcoefficients available to measure: κ₀₀, κ₀₁, and κ₁₁. The local patternof the Bragg grating determines the magnitude of the cross coupling orthe coupling coefficients. As a result, distributed measurements ofthese three coupling coefficients can be used to calculate the localfrequency of the grating (which determined by the tilt of the grating,which in turn determined by the amount of bending of the grating) atevery point along the fiber wave guide. The manner in which thesecoupling coefficients are measured, and the corrections used tocalculate the local frequency of a Bragg grating in the fiber core aredescribed in detail below for a non-limiting and example case of a threedimensional waveguide.

Previous fiber-optic shape sensing systems use a multi-core fiber ormultiple single core fibers, where each core is a single mode fiber andtypically has gratings written along the length of the core. Bend,twist, and/or strain cause local stretching or compressing of thegrating pattern. These grating changes relate directly to the appliedbend, twist, or strain and can be quantified by measuring the phasechange vs. distance along the fiber with respect to a reference state. Amatrix can be calculated via knowledge of the fiber geometry and/orcalibration, and used to convert phase change in four cores to bend,twist, and strain.

In this example case of a single core fiber with multiple modes, thevariations in the core grating pattern are measured and processed withdistance along the fiber and with cross-sectional location in the fibercore. For example, if the fiber is bent in one direction, the gratingpattern is compressed on the side of the core on the inside of the bend,stretched on the side of the core on the outside of the bend, andunchanged along the neutral axis. FIG. 7A shows a grating in an unbentmulti-core single mode fiber, and FIG. 7B shows the same multi-core whenbent. FIG. 7C shows a single core multiple mode fiber with the samegrating pattern bent in the same way. The periodicity of the gratingpattern varies across the cross-sectional area of the core.

FIG. 8 is a reference diagram of a single core fiber with x, y, z axes,a vector/radius r, and an angle θ. These labels are used in the some ofthe following figures and in the description below.

Although a single grating in the core may be used to determine strain,it is not sufficient by itself to determine another important shapeparameter—twist. Twist may be measured using a helixed fiber that usesmultiple single mode cores disposed in a helical manner as described inthe patents referenced in the introduction. But with a single core,multiple mode sensing fiber, a helixed fiber is no longer available formeasuring twist.

In various embodiments, a grating pattern can be used to detect twist inthe fiber. Construction of an example grating pattern is now described.FIG. 9 shows an example of a tilted Bragg grating written in a singlecore. FIG. 10 shows an example intensity plot of the index modulation ofan example tilted Bragg grating. An index modulation is another way torepresent the grating pattern. Note the x axis for the core is plottedhorizontally, and z axis for the core is plotted vertically.

This type of grating can be written in a fiber core using a phase maskthat is tilted with respect to the fiber. FIG. 11 shows an example phasemask to write a Bragg grating in a typical orientation with respect tothe fiber when writing gratings. FIG. 12 shows how the mask can betilted to write a tilted grating. Alternatively, the mask can be madewith the pattern tilted in order to write the desired tilted grating.

Then, a second tilted grating is written on the core as shown in theexample intensity plot of the index modulation or index change of FIG.13, but tilted in an opposing direction in the same location. The resultis constructive and destructive interference. FIG. 14 shows an exampleintensity plot of a grating pattern created when the two tilted gratingsare written overtop of one another in the core. The dim/blurry areasalong the vertical z axis at about 20, 60, 140, and 180 units ofdistance on the horizontal x axis represent areas where destructiveinterference has “washed out” the grating amplitude.

FIG. 15 shows an example amplitude envelope of the overlapped tiltedgratings. Lighter areas represent stronger grating amplitude, and darkerareas represent weaker grating amplitude where the grating amplitude iswashed out. This plot represents the grating amplitude along the lengthof the fiber (axis z) vs. one cross sectional axis of the fiber, x.

FIG. 16 is an example intensity plot of a germanium doping concentrationin an example fiber core looking at a cross section of the core in thex-y plane. This example fiber is only photosensitive in the core, andthe grating only exists in the core.

FIG. 17 shows the grating amplitude features created by writing tiltedgratings on top of one another within core 13 of the optical fiber 10.FIG. 18 is an example intensity plot of an end view of the fiber in thex-y plane created by writing tilted gratings on top of one another.

A grating pattern (like the pattern described above and an amplitudelike that shown in FIG. 15) written in a single core varies with bend,strain, and twist. Thus, various embodiments determine thecross-sectional variation in the grating pattern—which can also bedescribed as the cross-sectional index perturbation—as a function ofdistance along the fiber. If the cross-sectional variation in thegrating pattern can be determined, then the bend, strain, and twist asfunction of distance along the fiber can be determined.

Applying coupled-mode theory, the index perturbation in a fiber coredetermines the cross-coupling between forward traveling and backwardtraveling modes. In this application, the index perturbation is the coregrating. The core is probed with light having forward traveling modes,and the backward traveling modes created by the grating are detected.Because the input (forward traveling) modes may be determined inadvance, and because the output (backward traveling) modes can bemeasured and the coupling coefficients determined, mode coupling is usedthe example embodiments to determine the index perturbation (the stateof the grating such as pulled, compressed, bent, and/or twisted) thatwas present.

The following assumes that the single core guides three modes for adetailed example. However, a single core that guides more than threemodes may also be used.

A representation of the index perturbation (the core grating) across thecross section of the example fiber at any given location may bedetermined by measuring the coupling coefficients between the backscattered modes. Assume that the single core fiber supports three modesE₀ (circular), E_(H) (horizontal), and E_(V) (vertical), two of themdegenerate. The three modes correspond to three linear polarizationsLP₀₁ (circular), LP_(11x) (horizontal), LP_(11y) (vertical) as set forthhere:

$E_{0} = {{LP}_{01} = {e^{- \frac{r^{2}}{w_{0}}}e^{i\;\beta_{0}z}}}$$E_{H} = {{LP}_{11x} = {e^{- \frac{r^{2}}{w_{1}}}r\;\sin\mspace{14mu}\theta\; e^{i\;\beta_{1}z}}}$$E_{V} = {{LP}_{11y} = {e^{- \frac{r^{2}}{w_{1}}}r\mspace{14mu}\cos\mspace{14mu}\theta\; e^{i\;\beta_{1}z}}}$The variables r and θ and the z axis are shown in FIG. 8. The variable wis the Gaussian width of the mode and β is the propagation constant ofthe mode.

FIG. 19 shows scalar field plots of three lowest order LP modes E₀(circular), E_(H) (horizontal), and E_(V) (vertical).

Based on perturbation theory, (see, e.g., Fundamentals of OpticalWaveguides, Katsunari Okamoto, Elsevier 2006), the coupling factor,κ_(mn) between any forward propagating mode, E_(m) and any backwardpropagating mode E_(n) is given by the equation below.κ_(mn) =∫∫E _(m)(x, y)η(x, y)E* _(n)(x, y)e ^(i(β) ^(m) ^(+β) ^(n)^(−k)z) dx dyHere η(x, y)e^(−ikz) is the periodic index perturbation (the grating inthe core) within the optical fiber, β_(m) and β_(n) are the propagationconstants of the two non-degenerate modes, x and y are the crosssectional axes, and z is the dimension along the axis of the fiber.

Expressing this equation in polar coordinates:

$\kappa_{mn} = {\int\limits_{0}^{a}{\int\limits_{0}^{2\pi}{{E_{m}\left( {r,\theta} \right)}{\eta\left( {r,\theta} \right)}{E_{n}^{*}\left( {r,\theta} \right)}e^{{i{({\beta_{m} + \beta_{n} - k})}}z}{rdrd}\;{\theta.}}}}$

Six different decomposition functions, ξ_(mn)(r,θ) are available fordetermining η(r,θ):ξ_(mn)(r,θ)=E _(m)(r,θ)E* _(n)(r,θ)

FIG. 20 illustrates the six decomposition functions of the six crosscoupling possibilities for this example.

FIG. 21 shows decomposition functions for decomposing index perturbationfor this example.

By measuring the phase and amplitude of all six coupling coefficients,κ_(mn), the grating modulation (e.g., changes in the grating caused bystrain, stress, and/or twist), η(r,θ)e^(−ikz), may be reconstructedusing these coefficients, κ_(mn), as the weight factor on summation ofthe decomposition function ξ_(mn)(r,θ).η({umlaut over (p)})≈Σκ_(mn)ξ_(mn)(r,θ)=Σκ_(mn) E _(m)(r,θ)E* _(n)(r,θ).

This decomposition method of determining the distribution of a Bragggrating across the cross section of a single multimode core wassimulated for two tilted and overlapped Bragg gratings, such asdescribed above. The plots in FIG. 21 represent the grating amplitudeacross the core cross-section and show simulated, compound, tiltedgrating index profiles on the left and the grating profiles successfullyreconstructed from the overlap function decomposition on the right. Thetop two images are without fiber twist, the bottom two with twist. Thesimulation results show that the overlap integrals provide significantreconstruction capability and the lower right reconstruction shows thattwist of the grating structure can be determined.

Before measurements are taken with a single multimode core, a “baseline”or reference measurement with the fiber in a known orientation, in thiscase straight and untwisted, as performed. Thereafter, normalmeasurements may be made with fiber moved, e.g., bent and/or twisted. Aphase “deformation” δϕ is then calculated based upon the phase changebetween these baseline and normal measurements.δϕ(r,θ)=∠{η_(move)(r,θ)·η*_(base)(r,θ)}Or equivalently in the x, y coordinate system:δϕ(x, y)=∠{η_(move)(x, y)·η*_(base)(x, y)}

Between the base and the moved measurement for this example, there canbe a horizontal bend, b_(x), that will be a linear phase change as afunction of x, a vertical bend, b_(y), that will be a linear phasechange as a function of y, a stretch, ε, that will be constant changeover the surface, or a twist, τ, that will be a rotation of the entirecomplex (phase and amplitude) scattering cross section.

Estimates can be determined for each of these shape parameters b_(x),b_(y), ε, and τ. These estimates can essentially function as a shapeconversion matrix to convert from phase measurements to pitch, yaw,twist and strain.

ɛ = ∠∫∫η_(move)(x, y) ⋅ η_(base)^(*)(x, y)dAb_(x) = max ∫∫η_(move)(x, y) ⋅ η_(base)^(*)(x, y)e^(ib_(x))dAb_(y) = max ∫∫η_(move)(x, y) ⋅ η_(base)^(*)(x, y)e^(ib_(y))dA$\tau = {\max{\int\limits_{0}^{2\pi}{{{\eta_{move}\left( {r,\theta} \right)} \cdot {\eta_{base}^{*}\left( {r,{\theta + \tau}} \right)}}d\;\theta}}}$

With the bend, twist, and strain terms b_(x), b_(y), ε, and τ, the shapeof the fiber may be determined using three dimensional rotations andprojections described in U.S. Pat. Nos. 7,781,724 and 8,773,650identified in the introduction.

Measurement of Coupling Coefficients

In addition to using the measured coupling coefficients (κ_(mn)) todetermine the index perturbation in a single multimode core, someembodiments also measure the coupling coefficients. Some embodiments getthe energy from single mode fibers into and out of each of the differentmodes in a way that is power efficient (the detecting fibers aretypically coupled to photodiode detectors in an Optical Frequency DomainReflectometry (OFDR) system). As one option, a fused tapered coupler,tapering multiple single mode fibers and fusing them to differentlocations on the multimode core, may be an option if the tapering can bedone as the mode coupling was monitored in a 3×3 coupler. Another optionabuts a bundle of single mode cores up against the larger multimodecore. Some embodiments use this option when the cores are surrounded bylittle or no cladding. Another option is to image the multimode coredirectly onto a bundle of single mode fibers with a large magnification.Some embodiments use this option when the system can function with theinefficiencies resulting from light that may be captured by the claddingsurrounding each of the single mode fibers.

Some embodiments including this last option of directly imaging themultimode core reduce the inefficiencies by using a micro lens arraywith a sufficiently large fill-factor and a numerical aperture (NA) thatmatches the optical fiber with the multimode core more closely (e.g.,between 0.1 and 0.25 in some cases). An example of a bulk-optic designfor getting energy from each of the different modes into a correspondingsingle-mode detecting fiber in a way that is power efficient is shown inFIG. 22. The design may use a commercially-available microlens array 22and includes a fiber array 20 of three single core, single modedetecting fibers for directing light from each of the three modes oflight provided from a single multimode core 13 in a single coremultimode sensing fiber 10 into three single core, single mode detectingfibers in the fiber array 20. Multimode light from multimode core 13 isimaged by lens 24 to the microlens array 22.

FIG. 22 shows specific values for dimensions and other parameters thatare useful for specific embodiments, and are only examples and are notlimiting.

The optical coupling mechanism shown in FIG. 22 is included in FIG. 23.FIG. 23 shows an example system for interrogating a single coremultimode sensing fiber 10 with multiple modes. The interrogation systemis controlled by an OFDR controller 30 that includes one or morecomputers 32 coupled to (i) a laser controller 34 that controls atunable laser 36 through a range of frequencies or wavelengths, (ii) adisplay, and (iii) acquisition circuitry including photodiodes, analogto digital conversion circuitry, sampling circuitry. Because there arethree modes, there are three reference branches and three measurementbranches of different lengths coupled to three single mode fibers thatare coupled to three polarization beams splitters (PBSs) that convertthe interfered light for each mode into s and p polarizations fordetection by respective s and p photodiodes. The laser controller 34controls the tunable laser 36 to probe the single core multimode sensingfiber 10 with three separate single core fibers in the three measurementbranches, each with a unique delay corresponding to a different lengthof fiber L, L1, and L2. The laser light from each of the three inputfibers is coupled via the fiber array 20 through the microlens array 22and lens 24 into the single core 13, and the reflected light from eachof the three modes from the single core grating is delivered by themicrolens array 22 to its corresponding single core single modedetecting fiber in the fiber array 20 for detection and processing inthe OFDR controller 30.

If the delays of the three input fibers are larger than the total delayassociated with the single core multimode sensing fiber 10, then all ofthe cross coupled terms between input and output fibers of themeasurement interferometer will appear at different delays and on uniquedetectors. If the fiber lengths for each output fiber are the same, thenthe overall delay is determined by the input fiber in this example.

FIG. 24 are graphs that illustrate 9 coupling terms m11-m33. The threegraphs of amplitude vs. delay represent the amplitude of light detectedon each of the three detectors at the acquisition circuitry 38corresponding to the three single mode fibers in the fiber array 20. Thefirst subscript on the coupling terms, m, identifies the input fiber andthe second subscript identifies the output fiber (the detector thesignal is detected on). So m31 represents the amount of light sent in onfiber 3 and detected on detector fiber 1.

Extraction of Fiber Modal Coupling From Instrumentation Coefficients

As demonstrated above, the index perturbation (grating) may bedetermined by measuring the cross coupling coefficients detected at themultiple (three in the example) single mode fibers in the fiber array 20that correspond to but are not identical to the modes of the single core13 in the single core multimode sensing fiber 10. FIG. 24 describedabove illustrates the coupling coefficients between the three measuredcomponents coupled to the single mode fibers in the fiber array 20.These cross-coupling coefficients for these single mode fibers can bemeasured directly by the OFDR controller 30, but they need to beconnected or converted to the actual modal cross-coupling coefficientsfor the three modes of the single core 13 in the single core multimodesensing fiber 10.

Some embodiments make this connection or conversion using a matrix, α,that describes how the light from the three input fibers in the fiberarray 20 couples into the three modes supported in the single multimodecore 13 of the single core multimode sensing fiber 10. FIG. 25 shows anexample model of an optical connection system as two matrices connectedby the alpha matrix α. FIG. 25 is represented using the mathematicalexpression below:

$\begin{bmatrix}A_{1} \\A_{2} \\A_{3}\end{bmatrix} = {{\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}\begin{bmatrix}E_{V} \\E_{0} \\E_{H}\end{bmatrix}} = {{\overset{\rightharpoonup}{A}}_{out} = {\overset{\_}{\overset{\_}{\alpha}}\overset{\rightharpoonup}{E}}}}$where Ā_(out) is the vector formed by the fields in each of thesingle-mode input/probe fibers in the fiber array 20, Ē is the vectorcomposed of the field in each of the three fiber modes in the singlecore 13, and α is the connecting matrix that couples the multimode fibermodes Ē to the single mode fiber fields Ā_(out). E₀ is the circularlysymmetric LP01 mode, E_(V) is the vertically oriented LP11 mode, andE_(H) is the horizontally oriented LP11 mode. Light can travel in bothdirections, and the conversion from light in the individual single modecores in the fiber array 20 to the light in the modes of the single coremultimode sensing fiber 10 are given by,

${\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}^{- 1}\begin{bmatrix}A_{1} \\A_{2} \\A_{3}\end{bmatrix}} = {\begin{bmatrix}E_{V} \\E_{0} \\E_{H}\end{bmatrix} = {{{\overset{\_}{\overset{\_}{\alpha}}}^{- 1}{\overset{\rightharpoonup}{A}}_{out}} = \overset{\rightharpoonup}{E}}}$

The expression is now written for the light travelling from thesingle-mode fibers in the fiber array 20, through the optics (microlensarray 22 and lens 24) into the single core multimode sensing fiber 10,then being coupled into the backward travelling mode, and then backthrough the optics (microlens array 22 and lens 24) and into thesingle-mode fibers in the fiber array 20 as follows:

${\overset{\rightharpoonup}{A}}_{out} = {{{\overset{\_}{\overset{\_}{\alpha\; K\;\alpha}}}^{- 1}{\overset{\rightharpoonup}{A}}_{in}} = {{{\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}\begin{bmatrix}\kappa_{11} & \kappa_{21} & \kappa_{31} \\\kappa_{21} & \kappa_{22} & \kappa_{32} \\\kappa_{31} & \kappa_{32} & \kappa_{33}\end{bmatrix}}\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}}^{- 1}\begin{bmatrix}A_{1} \\A_{2} \\A_{3}\end{bmatrix}}}$

The combination of the above three matrices, αKα ⁻¹, forms the overallcoupling coefficients between input fibers and output fibers shown inFIG. 24 and is referred to hereafter as the M matrix.

${\overset{\rightharpoonup}{A}}_{out} = {{{\overset{\_}{\overset{\_}{\alpha\; K\;\alpha}}}^{- 1}{\overset{\rightharpoonup}{A}}_{in}} = {{\begin{bmatrix}m_{11} & m_{21} & m_{31} \\m_{21} & m_{22} & m_{32} \\m_{31} & m_{32} & m_{33}\end{bmatrix}\begin{bmatrix}A_{1} \\A_{2} \\A_{3}\end{bmatrix}} = {\overset{\_}{\overset{\_}{M}}{\overset{\rightharpoonup}{A}}_{in}}}}$

In this example, αKα ⁻¹ is what is measured with the time-delayedinterrogation network shown in FIG. 23. Although the objective is todetermine the modal cross-coupling coefficients represented by thecross-coupling coefficients matrix K corresponding to the state of thecore grating, what can be actually measured by the OFDR interrogationsystem is the M matrix.

Embodiments determine the elements that form the matrix M in thefollowing manner. An OFDR scan of a single core multimode fiber with acleaved end is used to separate out the E₀ (LP01) mode based uponpropagation time to the end of the cleave and also to separate the crosscoupling terms. FIG. 26 is a graph of an example impulse response orpropagation delay of a single core multimode fiber with a cleaved end.The reflection from the cleaved end is shown and labeled as “fiberendface.” The different propagation modes propagate along the fiber atdifferent speeds, which means they can be separated and identified. Forexample, because the E₀ (LP01) mode has a longer propagation time, itsreflection will arrive later at the OFDR detector than the E_(V) andE_(H) (LP11) mode reflections. The light that travels down the fiber inthe E₀ (LP01) mode and couples to the E_(V) and E_(H) (LP11) modes atthe cleave will arrive with a delay exactly in between the two modes.

FIG. 27 is a graph of the example impulse response delays in FIG. 26relabeled to identify the modal assignments (coupling coefficients).From FIG. 27, the three groups of coupling coefficients can be separatedand filtered around each of the different delays (the distinct andseparate impulse response amplitude lines). The terms are isolated byfirst windowing the complex data around the reflection peak, and thenperforming a Fourier transform to compute the frequency domain complexspectrum of the peak. First, the reflected coupling coefficient κ₂₂ formode E₀ (LP01) is isolated using the windowing and transformation tomathematically separate the peak on all channels and at all offsetdelays to give a matrix of filtered values.

$\overset{\_}{\overset{\_}{M_{0}}} = {{\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}\begin{bmatrix}0 & 0 & 0 \\0 & \kappa_{22} & 0 \\0 & 0 & 0\end{bmatrix}}\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}}^{- 1}$To implement this isolation in this example, the OFDR controller appliesa time domain window filter (like a bandpass filter) around the E₀(LP01) delay peak.

Next, the light that travels exclusively in the E_(V) and E_(H) (LP11)modes and corresponding to the following is isolated:

$\overset{\_}{\overset{\_}{M_{1}}} = {{\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}\begin{bmatrix}\kappa_{11} & 0 & \kappa_{31} \\0 & 0 & 0 \\\kappa_{13} & 0 & \kappa_{33}\end{bmatrix}}\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}}^{- 1}$

And finally the light that crosses between E₀ (LP01) and the E_(V) andE_(H) (LP11) modes at the cleave is isolated as follows:

$\overset{\_}{\overset{\_}{M_{2}}} = {{\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}\begin{bmatrix}0 & \kappa_{21} & 0 \\\kappa_{12} & 0 & \kappa_{32} \\0 & \kappa_{23} & 0\end{bmatrix}}\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}}^{- 1}$

Next, assuming a flat, 0 degree cleave at the end of the fiber, thecross coupling terms (κ₁₃ and κ₃₁) are driven to zero because theperturbations function η(r,θ) for a zero degree cleave is a constant,giving a simplified matrix:

$\overset{\_}{\overset{\_}{M_{1}}} = {{\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}\begin{bmatrix}\kappa_{11} & 0 & 0 \\0 & 0 & 0 \\0 & 0 & \kappa_{33}\end{bmatrix}}\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}}^{- 1}$

Adding M₀ and M₁ together gives the following diagonal matrix betweenthe input and output coupling matrices.

${\overset{\_}{\overset{\_}{M_{0}}} + \overset{\_}{\overset{\_}{M_{1}}}} = {{\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}\begin{bmatrix}\kappa_{11} & 0 & 0 \\0 & \kappa_{22} & 0 \\0 & 0 & \kappa_{33}\end{bmatrix}}\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}}^{- 1}$which can be rewritten as by moving the inverse matrix to the left sideof the equality:

${\overset{\_}{\overset{\_}{M_{0}}} + {\overset{\_}{\overset{\_}{M_{1}}}\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}}} = {\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}\begin{bmatrix}\kappa_{11} & 0 & 0 \\0 & \kappa_{22} & 0 \\0 & 0 & \kappa_{33}\end{bmatrix}}$

This last equation takes the form of an eigenvalue problem:AS=SΛwhere S is a matrix whose columns are the eigenvectors of A and Λ is adiagonal matrix whose elements are the eigenvalues. Since matrix A canbe measured by the OFDR interrogation system, the coupling coefficients(κ₁₁, κ₂₂, and κ₃₃) can be determined.

Finding the matrices that convert the measurements, M₀ +M₁ , to the modecoupling coefficients (κ₁₁, κ₂₂, and κ₃₃) is now an eigenvalue problemwhere the coupling coefficients (κ₁₁, κ₂₂, and κ₃₃) are the eigenvaluesof the matrix, M₀ +M₁ .

The connector matrix:

$\begin{bmatrix}\alpha_{11} & \alpha_{21} & \alpha_{31} \\\alpha_{21} & \alpha_{22} & \alpha_{32} \\\alpha_{31} & \alpha_{32} & \alpha_{33}\end{bmatrix}\quad$can be constructed from the eigenvectors of M₀ +M₁ using linear algebra.

Once the connector α matrix, which defines coupling between the singlemode fibers and the single core multimode sensing fiber, is determined,it remains constant for a given fiber connection. This connector αmatrix and its inverse define how light couples from the OFDRinterrogation system to the modes of the single core multimode sensingfiber and back. Knowing the values for this connector α matrix and theinput light, the output light is measured by the OFDR interrogationsystem, and from these three things, the OFDR interrogation systemcalculates the values of the current modal cross coupling coefficients,or the K matrix. Based on these modal cross coupling coefficient values,the phase perturbations in the single core multimode sensing fiber 10that produced the measured coupling (κ) coefficients and which representthe state of the grating in the single core multimode sensing fiber 10are calculated. From these determined phase perturbations correspondingto the state of the core grating, the bend and twist of the single coremultimode sensing fiber 10 can be calculated at every point along thesingle core multimode fiber 10. These bend and twist values can then beused to calculate the shape of the multimode sensing fiber 10.

Accounting for Group Delay Differences Between Modes

The approach described above provides measurements of the couplingbetween modes along the length of the single core multimode sensingfiber 10 as a function of time delay. In some embodiments, furthercorrections are made to address differing group velocities betweenmodes. For example, to address the differing group velocities causingthe mode coupling coefficients (κ) to be spread over different delayranges depending on which modes the light propagated in and for howlong.

FIG. 28 graphs example fiber mode responses with exaggerated time-lengthdifferences to illustrate how the effective length of the fiber appearsto be different for each of the modes due to differences in group delaybetween the modes. In FIG. 24, the time delay amounts are the same,while in FIG. 28, the time delay amounts are different due to groupdelay effects. These group delay effects can be accounted for byresampling the data when the coupling coefficients are identified. Insome embodiments, the coupling coefficients identification includesmapping the physical location in the fiber where the coupling took placeto the same index for each mode. Here, the term index is the locationalong the length of the fiber before it has been scaled to engineeringlengths. The data is lined up to represent the same location at the sameindex. This mapping is performed for each of the 6 independent couplingcoefficients in this example (although there are 9 couplingcoefficients, several have the same effective group index such as κ₁₃and κ₃₁).

FIG. 29 is a graph showing an example of resampling and alignment ofcoupling coefficients κ₁₁, κ₂₁, and κ₃₁ showing conversion of them fromthe time domain to the spatial domain (the horizontal axis in FIG. 29aligns with the z axis of the single core multimode sensing fiber 10.This conversion is accomplished by multiplying the time label of each ofthe coupling coefficients κ₁₁, κ₂₁, and κ₃₁ in the time domain by thespeed of light for that coupling coefficient.

In addition to propagating with different group delays, each mode alsopropagates with a different wavenumber, or effective refractive index,meaning that each mode accumulates phase at a different rate as itpropagates down the single core multimode sensing fiber 10. FIG. 30graphs example phase accumulations for different coupling coefficientsas a function of delay.

The LP11 and LP01 self-coupling terms are both weighted integrals overthe single core multimode sensing fiber 10 in this example. As a result,even for random Rayleigh scatter, the LP11 and LP01 self-coupling termsare substantially similar. If the sensing fiber is straight, this commonterm in the coupling terms allows precise measurements of the differencebetween the phase propagation terms for the LP11 and LP01 modes alongthe fiber length. Further, this difference in effective refractive indexcan be used to predict the phase change in the cross term phases, suchas by assuming the cross term phase accumulations are exactly halfwaybetween the pure LP11 and LP01 phase delays. These calculated phasechanges due to the different effective refractive indexes of the modesare then applied to the measured and resampled coupling coefficientsbefore the scattering cross section is calculated.

FIG. 31 is a flowchart showing example procedures for using a singlecore, multiple mode fiber for sensing shape in accordance with exampleembodiments. In step S1, light is input into three (or more) modessupported by the single multimode core fiber. The grating of the corecauses forward traveling modes to be reflected into backward travelingmodes as shown in step S2. In step S3, reflected light is detected,converted into electrical signals, converted from analog format intodigital format, processed in the OFDR controller to calculate the crosscoupling coefficients. In step S4, decomposition functions are used todetermine the index perturbation, or the current state of the grating,when compared to a measurement in a known state (such as the fiberpositioned in a straight line). At step S5, the pitch, yaw, twist, andstrain from the change in the index perturbation is determined. Then, instep S6, the fiber shape is determined from the pitch, yaw, twist, andstrain determined in step S5.

FIG. 32 is a flowchart showing example procedures for using a singlecore, multiple mode fiber for sensing shape in accordance with exampleembodiments. In step S10, light from the tunable laser is split betweenmeasurement and reference paths using for example the OFDR system inFIG. 23. In step S11, the measurement light is split into three inputfibers, each input fiber having a different delay. In step S12, lightfrom the three input fibers is coupled into the single core multimodesensing fiber using a lens array. In step S13, forward propagating lifefrom the three modes is coupled into backward traveling light in thethree modes via the core grating. In step S14, the light in the backwardtraveling modes is coupled into the three single mode fibers (now outputfibers) via the lens array. In step S15, this backward traveling lightis combined with the reference light and detected on three detectors. Instep S16, data from the three detectors is processed to determine the Mmatrix which couples the coupling coefficients between the input lightand the output light. In step S17, using the alpha matrix, which is thematrix describing the coupling between the single mode input fibers andthe modes of the single core multimode fiber, cross couplingcoefficients between the modes of the single core multimode fiber aredetermined. These cross coupling coefficients form the K matrix. In stepS18, these cross coupling coefficients are then used to calculate thecurrent index perturbation, which is the core grating, and from there,the shape of the single core multimode sensing fiber is calculated as insteps S4-S6 in FIG. 31.

FIG. 33 is a flowchart showing example procedures for calibrating andthen using a single core, multiple mode fiber for sensing shape inaccordance with example embodiments. Two different process stages areincluded and coupled loosely with a dashed line. The first process stageis performed before the fiber is used for shape measurements. The secondprocess stage is used when the sensing fiber is to perform shapemeasurement.

In step S20, the single core multimode fiber is connected to a multimodeinterrogation system such as the OFDR system shown in FIG. 23. In stepS21, the shape sensing fiber is positioned in a straight line or inanother known orientation. The cleaved end response of the fiber ismeasured in step S22. Then, the decoupling matrix is calculated toconvert the measurement modes to fiber modes in step S23. The decouplingmatrix is referred to above as the alpha matrix and is determinedfollowing steps S20-S23. The next steps S24 through S26 relate todetermining reference or baseline measurements of the shape sensingfiber in a known state. In step S24, the cleaved end is terminated, andthe coupling coefficients are measured for the shape sensing fiber. Instep S25, the coupling coefficients are resampled into the same physicalframe. Then, in step S26, corrections are determined for the wavenumberdifferences.

Moving over to the next process stage of the flowchart, step S30describes changing the shape of the fiber to a measurement orientation.In step S31, the fiber response is measured using the interrogationsystem. Then, the decoupling or alpha matrix is applied to the measuredresponse in step S32. The coupling coefficients are resampled in stepS33, and the wavenumber corrections are applied in step S34. The indexmodulation profile change is calculated at each point along the sensingfiber in step S35. In step S36, the bend, twist, and strain arecalculated each point along the fiber, and from these values, the shapeof the sensing fiber is calculated in step S37.

Example Multimode Single Core Fiber Sensor Twist Sensitivity

Having determined how to make and use a single core multimode fiber todetermine shape using an OFDR interrogation method and overlapped tiltedgratings, the inventors tested different fiber core sizes and estimatedtheir sensitivity to detecting twist.

An example fiber that admits only three modes was designed and thenchecked to verify that this fiber provides reasonable twist sensitivity.The number of modes in a step-index multimode fiber is described by anormalized frequency parameter known as the V-number. A V-number ofaround 3.5 was selected in order to guide the LP11 modes tightly, whileexcluding the LP21 and LP02 modes.

FIG. 36 is a graph showing modal propagation values for LinearPolarization Modes vs. V-number. The V-number is given by the equationbelow.

$V = {{\frac{2\pi\; r}{\lambda}\sqrt{n_{1}^{2} - n_{2}^{2}}} = {\frac{2\pi\; r}{\lambda}{NA}}}$where n₁ is the index of refraction of the core, n₂ is the index of thecladding, λ is the wavelength of the light, r is the core radius, and NAis the Numerical Aperture. Using V=3.5, the selected operatingwavelength, and a selected Numerical Aperture permits calculation of thecore radius as follows:

$3.5 = {\frac{2\pi\; r}{1540 \times 10^{- 9}}0.2}$Solving for r yields:

$\frac{3.5\left( {1540 \times 10^{- 9}} \right)}{{0.2 \cdot 2}\pi} = {r = {4.3 \times {10^{- 6}.}}}$

The rotational period of the skew rays (e.g., as shown in FIG. 34A) canbe calculated from the difference in the wavenumbers between the modespresent in the fiber. This difference may be characterized by theparameter b, which can be used to calculate the wavenumbers for eachmode. FIG. 37 is a graph showing a normalized propagation constant Bversus normalized frequency V for TE and TM modes. This graph is used toidentify values for b at the selected V number of 3.5 as b₀=0.34 andb₁=0.75. The definition of the b number is

${b = \frac{\beta^{2} - k_{2}^{2}}{k_{1}^{2} - k_{2}^{2}}},$where β is the propagation constant and k is the wavenumber. Thedefinition of the wave number is

$k_{1} = {\frac{2\pi}{\lambda}{n_{1}.}}$

The propagation constant β as a function of b is solved in accordancewith:√{square root over (b(k ₁ ² −k ₂ ²)+k ₂ ²)}=βFactoring out the vacuum wavenumber, 2π/λ, collecting terms, andapplying the definition of Numerical Aperture (NA) in an optical fiber,NA=√{square root over (n₁ ²−n₂ ²)}, this equation may be expressed as:

${\frac{2\pi}{\lambda}{NA}\sqrt{b + \frac{n_{2}^{2}}{{NA}^{2}}}} = {\beta.}$

An expression for the beat length between the two non-degenerate modesis:

${\frac{2\pi}{\lambda}{{NA}\left\lbrack {\sqrt{b_{1} + \frac{n_{2}^{2}}{{NA}^{2}}} - \sqrt{b_{0} + \frac{n_{2}^{2}}{{NA}^{2}}}} \right\rbrack}} = {\beta_{1} - {\beta_{0}.}}$

Pulling the common terms out from under the radicals and factoring outthe common terms results in:

${\frac{2\pi}{\lambda}{n_{2}\left\lbrack {\sqrt{1 + {\frac{b_{1}}{n_{2}^{2}}{NA}^{2}}} - \sqrt{1 + {\frac{b_{0}}{n_{2}^{2}}{NA}^{2}}}} \right\rbrack}} = {\beta_{1} - {\beta_{0}.}}$

A binomial approximation is applied to the terms under the radicals, andterms are canceled and common factors removed to yield:

${\frac{\pi}{\lambda}{\frac{{NA}^{2}}{n_{2}}\left\lbrack {b_{1} - b_{0}} \right\rbrack}} = {{\beta_{1} - \beta_{0}} = {{\Delta\beta}.}}$

Substituting numerical values gives:

${\frac{\pi}{1540 \times 10^{- 9}}{\frac{0.2^{2}}{1.47}\left\lbrack {0.75 - 0.34} \right\rbrack}} = {{\left( {55\text{,}510} \right)(0.41)} = {{23 \times 10^{3}} = {{\Delta\beta}.}}}$Using

$L = \frac{2\pi}{\Delta\beta}$gives a beat length of 270 microns.

Estimating twist sensitivity using an approximate radius for the higherorder modes of r=0.002 mm produces:

$\frac{r^{2}}{L} = {\frac{\left( {0.002\mspace{14mu}{mm}} \right)^{2}}{0.270} = {14 \times 10^{- 6}\mspace{14mu}{{mm}.}}}$

This twist sensitivity may be doubled to a value that is about half ofthe sensitivity in a multi-core shape sensing fiber. This demonstratesthat a simple step-index, three-mode single core fiber has sufficienttwist sensitivity to function as an effective shape sensor.

Different shape cores with different numbers of multimodes may be used.For example, a single small core as shown in FIG. 35 supports just a few(e.g., 3) modes, whereas a single large core as shown in FIG. 38supports many (e.g., 200) modes in addition to the axial ray (shown inFIG. 34B) and skew rays (shown in FIG. 34A) described above. Further,all of these modes can couple together when the fiber is perturbed,leading to a multitude of interactions that are measured in order to getaccurate results. This large number of measurements may or may not beworthwhile depending on the application.

FIG. 39 shows an example single core multimode fiber where the core isannular and supports 30 modes. Additional sensitivity afforded by thelarger core shown in FIG. 38 may be achieved using an annular core andeliminate the interior modes. Recognizing that axial rays can actuallypropagate parallel to the axis, but not on the axis, the annular corelimits the number of modes present and the complexity of themeasurement.

The technology described provides a single core multimode fiber that canbe used to accurately sense shape and can be manufactured quite simplyand cost effectively as compared to multicore shape sensing fiber.

The technology described above also has wide and diverse applications.One non-limiting example application is to a fiber optic shape sensingsystem for a robotic surgical arm in which one or more of the varioustechnical features and/or embodiments described above may be used.

The techniques described herein can be implemented using a controlsystem including at least one memory and at least one processor, andoften a plurality of processors. The control system also includesprogrammed instructions (e.g., a computer-readable medium storing theinstructions) to implement some or all of the methods described inaccordance with aspects disclosed herein. The control system may includetwo or more data processing circuits with one portion of the processingoptionally being performed on or adjacent the tool, and another portionof the processing being performed at a station (e.g. an operator inputsystem or central processing system or the like) remote from the tool.Any of a wide variety of centralized or distributed data processingarchitectures may be employed. Similarly, the programmed instructionsmay be implemented as a number of separate programs or subroutines, orthey may be integrated into a number of other aspects of theteleoperational systems described herein. In one embodiment, the controlsystem supports wireless communication protocols such as Bluetooth,IrDA, HomeRF, IEEE 802.11, DECT, and Wireless Telemetry.

Although various embodiments have been shown and described in detail,the claims are not limited to any particular embodiment or example. Noneof the above description should be read as implying that any particularelement, step, range, or function is essential such that it must beincluded in the claims scope. The scope of patented subject matter isdefined only by the claims. The extent of legal protection is defined bythe words recited in the allowed claims and their equivalents. Allstructural and functional equivalents to the elements of theabove-described preferred embodiment that are known to those of ordinaryskill in the art are expressly incorporated herein by reference and areintended to be encompassed by the present claims. Moreover, it is notnecessary for a device or method to address each and every problemsought to be solved by the technology described, for it to beencompassed by the present claims. No claim is intended to invoke 35 USC§ 112(f) unless the words “means for” or “step for” are used.Furthermore, no embodiment, feature, component, or step in thisspecification is intended to be dedicated to the public regardless ofwhether the embodiment, feature, component, or step is recited in theclaims.

We claim:
 1. An optical interrogation system for a sensing fiber, thesystem comprising: interferometric apparatus configured to probe asingle multimode core of the sensing fiber over a range of predeterminedwavelengths, and to detect measurement interferometric data associatedwith multiple light propagating modes of the single multimode core foreach predetermined wavelength in the range, the interferometricapparatus comprising multiple interferometers, the multipleinterferometers including multiple reference branches and multiplemeasurement branches, the multiple measurement branches comprising anarray of single core, single-mode fibers configured to be couple to themultimode core of the sensing fiber; and data processing circuitryconfigured to process the measurement interferometric data associatedwith the multiple light propagating modes of the single core todetermine one or more shape-sensing parameters of the sensing fiber. 2.The optical interrogation system of claim 1, wherein the one or moreshape-sensing parameters include at least one parameter selected fromthe group consisting of strain, bend, and twist parameters, and whereinthe data processing circuitry is further configured to determine a shapeof the sensing fiber based on the one or more shape-sensing parameters.3. The optical interrogation system of claim 1, wherein the dataprocessing circuitry is configured to process the measurementinterferometric data to determine the one or more shape-sensingparameters by: processing the measurement interferometric data todetermine a variation in a cross-sectional refractive index perturbationof the single multimode core, wherein the variation varies with the oneor more shape-sensing parameters; and, determining the one or moreshape-sensing parameters based at least in part on the variation.
 4. Theoptical interrogation system of claim 3, wherein processing themeasurement interferometric data to determine the variation in thecross-sectional refractive index perturbation comprises: processing themeasurement interferometric data to determine modal couplingcoefficients between the multiple light propagating modes; anddetermining the cross-sectional index perturbation based on the modalcoupling coefficients.
 5. The optical interrogation system of claim 4,wherein processing the measurement interferometric data to determine themodal coupling coefficients comprises: processing the measurementinterferometric data to determine coupling terms between light inputinto the single multimode core and reflected light output from thesingle multimode core; and determining the modal coupling coefficientsbased on the coupling terms.
 6. The optical interrogation system ofclaim 5, wherein different cores of the array of single core,single-mode fibers have different optical delays, and wherein thedifferent optical delays cause, for each core of the array of singlecore, single-mode fibers, signals associated with different couplingterms to appear at different delays within the measurementinterferometric data.
 7. The optical interrogation system of claim 3,wherein the cross-sectional refractive index perturbation of the singlemultimode core comprises a periodic refractive index perturbationcorresponding to one or more fiber Bragg gratings in the singlemultimode core.
 8. The optical interrogation system of claim 7, whereinthe one or more fiber Bragg gratings comprise overlapping tiltedgratings.
 9. The optical interrogation system of claim 1, wherein theinterferometric apparatus further includes: an optical couplingmechanism comprising a microlens array, the optical coupling mechanismconfigured to image light from the single multimode core to the array ofsingle core, single-mode fibers.
 10. The optical interrogation system ofclaim 9, wherein the optical coupling mechanism further comprises a lensconfigured to image the light from the single multimode core to themicrolens array.
 11. The optical interrogation system in claim 1,further comprising the sensing fiber, wherein the single core,single-mode fibers are tapered and fused to different locations on thesingle multimode core of the sensing fiber.
 12. The opticalinterrogation system in claim 1, wherein the single core, single-modefibers are configured to receive light from the single multimode core byabutting against the single multimode core.
 13. A method of operating anoptical interrogation system to sense with a sensing fiber coupled to aninterferometric apparatus of the optical interrogation system, theinterferometric apparatus comprising multiple interferometers, themultiple interferometers comprising multiple reference branches andmultiple measurement branches, and the sensing fiber comprising a singlemultimode core, the method comprising: probing, with the interferometricapparatus and through an array of single core, single-mode fibers of themultiple measurement branches, the single multimode core over a range ofpredetermined wavelengths; detecting, with the interferometricapparatus, measurement interferometric data associated with multiplelight propagating modes of the single multimode core for eachpredetermined wavelength in the range; and processing, with a dataprocessor, the measurement interferometric data associated with themultiple light propagating modes of the single multimode core todetermine one or more shape-sensing parameters of the sensing fiber. 14.The method of claim 13, wherein processing the measurementinterferometric data comprises: processing the measurementinterferometric data to determine a variation in a cross-sectionalrefractive index perturbation of the single multimode core, wherein thevariation varies with the one or more shape-sensing parameters; anddetermining the one or more shape-sensing parameters based on thevariation in the cross-sectional refractive index perturbation.
 15. Themethod of claim 14, wherein processing the measurement interferometricdata to determine the variation in the cross-sectional refractive indexperturbation comprises: determining modal coupling coefficients betweenthe multiple light propagating modes; and determining thecross-sectional index perturbation based on the modal couplingcoefficients.
 16. The method of claim 15, wherein determining the modalcoupling coefficients comprises: determining coupling terms betweenlight input into the single multimode core and reflected light outputfrom the single multimode core; and determining the modal coupling basedon the coupling terms.
 17. The method of claim 13, further comprising:imaging, with an optical coupling mechanism, light from the singlemultimode core to the array of single core, single mode fibers, whereinthe optical coupling mechanism comprises a microlens array.
 18. Themethod of claim 17, wherein imaging light from the single multimode coreto the array of single core, single mode fibers comprises: imaging, witha lens, the light from the single multimode core to the microlens array.19. A non-transitory machine-readable medium comprising a plurality ofmachine-readable instructions which, when executed by one or moreprocessors associated with an optical interrogation system comprisingmultiple interferometers, the multiple interferometers comprisingmultiple reference branches and multiple measurement branches, areadapted to cause the one or more processors to perform operationscomprising: operating the optical interrogation system to probe, throughan array of single core, single-mode fibers of the multiple measurementbranches, a single multimode core of a sensing fiber over a range ofwavelengths; operating the optical interrogation system to detectmeasurement interferometric data associated with multiple lightpropagating modes of the single multimode core for each wavelength ofthe range of wavelengths; and processing the measurement interferometricdata associated with the multiple light propagating modes of the singlemultimode core to determine one or more shape-sensing parameters of thesensing fiber.
 20. The machine-readable medium of claim 19, whereinprocessing the measurement interferometric data comprises: determiningthe one or more shape-sensing parameters based on a variation of across-sectional refractive index perturbation of the single multimodecore.
 21. The machine-readable medium of claim 20, wherein determiningthe one or more shape-sensing parameters based on a variation of across-sectional refractive index perturbation of the single multimodecore comprises: determining coupling terms between light input into thesingle multimode core from the array of single core, single-mode fibersand reflected light output from the single multimode core to the arrayof single core, single-mode fibers; determining, based on the couplingterms, modal coupling coefficients between multiple light propagatingmodes of the single multimode core; determining, based on the modalcoupling coefficients, the cross-sectional refractive indexperturbation; determining the variation of the cross-sectionalrefractive index perturbation relative to a reference cross-sectionalrefractive index perturbation; and determining, based on the variation,the one or more shape-sensing parameters.